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( ) 2014 2014 1/119

= (ISS) ISS ISS ISS iss-clf iss-clf ISS = (ISS) FB 2014 2/119

= (ISS) ISS ISS ISS iss-clf iss-clf ISS R + : 0 K: γ: R + R + K γ γ(0) = 0 K : γ: R + R + K γ K γ(r) (r ) FB K K K K R + R + 2014 3/119

( ) = (ISS) ISS ISS ISS iss-clf iss-clf ISS KL: β: R + R + R + KL s β(,s) K r β(r, ) β(r, s) 0, (s ) r FB 1 0.75 0.5 0.25 0 0.2 0.4 0.6 0.8 1 0.6 0.4 0.8 s 1 0 0.2 2014 4/119

= (ISS) = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB =(Input-to-State Stability,ISS): ẋ = f(x, t)+g 1 (x, t)d x =0 t = KL β K χ t 0 (> 0) x(t 0 ) [0, ) d( ) ( ) x(t) β( x(t 0 ),t t 0 )+χ sup t 0 τ t d(τ), t>t 0 β χ d 2014 5/119

= (ISS) ISS ISS ISS iss-clf iss-clf ISS FB ẋ = Ax + Bd x(t) = exp(a(t t 0 ))x(t 0 )+ t t 0 exp(a(t τ))bd(τ)dτ x(t) exp{(max Re(λ i (A)))(t t 0 )} x(t 0 ) + i t 0 t 0 τ t exp(a(τ))b dτ sup d(τ) A ISS 2014 6/119

ISS = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB ISS (Sontag and Wang 1995): ẋ = f(x, t)+g 1 (x, t)d f(0,t)=0 V : R n R + R + γ 1 ( x ) V (x, t) γ 2 ( x ) V = V t + V x f(x, t)+ V x g 1(x, t)d γ 3 ( x ), for x ρ( d ) γ 1, γ 2, ρ K γ 3 K ISS χ = γ 1 1 γ 2 ρ 2014 7/119

ISS = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB γ 1 ( x ) V (x) γ 2 ( x ) V ISS (Sontag and Wang 1996): ẋ = f(x)+g 1 (x)d f(0) = 0 V : R n R + V a( x )+b( d ) a( ), b( ) K ISS d a( ) K x V < 0 2014 8/119

- = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB - (ISS) x(t) β( x(t 0 ),t t 0 )+χ( d ) - (IISS) α( x(t) ) β( x(t 0 ),t t 0 )+ - t t t 0 α( x(τ) )dτ β( x(t 0 ),t t 0 )+ t 0 χ( d(τ) )dτ t t 0 χ( d(τ) )dτ - (ISS) - ISS IISS 2014 9/119

ISS = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB : ẋ = f(x)+g 1 (x)d + g 2 (x)u d u : u = α(x) ẋ = f(x)+g 1 (x)d = {f(x)+g 2 (x)α(x)} + g 1 (x)d ISS α(x) 2014 10 / 119

iss-clf = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB ISS-control Lyapunov function (iss-clf): V iss-clf K ρ d x ρ( d ) inf u R m{l f V + L g1 Vd+ L g2 Vu} < 0, x 0 ISS u V (x) 2014 11 / 119

iss-clf = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB Iss-clf : V iss-clf K ρ L g2 V (x) =0 L f V (x)+ L g1 V (x) ρ 1 ( x ) < 0, x 0 : d = {ρ 1 ( x )/ L g1 V }(L g1 V ) T : x ρ( d ) inf {L f V + L g1 Vd+ L g2 Vu} inf {L f V + L g1 V d + L g2 Vu} u u inf {L f V + L g1 V ρ 1 ( x )+L g2 Vu} < 0, x 0 u 2014 12 / 119

iss = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB u g 2 (x) =0 ISS-clf 3 ISS : V K ρ L f V (x)+ L g1 V (x) ρ 1 ( x ) < 0, x 0 ISS 2014 13 / 119

ISS = (ISS) ISS ISS ISS iss-clf iss-clf ISS FB ẋ = f(x)+g 1 (x)d + g 2 (x)u ISS u = α(x) scp iss-clf : Sontag and Wang : Sontag-type () ISS Sontag-type u = α(x) ω + ω 2 +(L g2 V (L g2 V ) T ) 2 = L g2 V (L g2 V ) T (L g2 V ) T, L g2 V 0 0, L g2 V =0 ω = L f V + L g1 V ρ 1 ( x ) 2014 14 / 119

ISS ( ) = (ISS) ISS ISS ISS iss-clf iss-clf ISS Sontag-type V L g2 V 0 : x ρ( d ) V L g2 V ( d ρ 1 ( x )) ω 2 +(L g2 V (L g2 V ) T ) 2 < 0 FB L g2 V =0 : Iss-clf x ρ( d ) V L f V + L g2 V ρ 1 ( x ) < 0 ISS scp Sontag-type 2014 15 / 119

= (ISS) FB 2014 16 / 119

(1) = (ISS) FB : J = F (x) min : x R n : g(x) =0, g(x) R m : h i (x) 0, i =1,...,l (h(x) 0 ) Lagrange : J ext (x, λ, μ, γ) =F (x)+λ T g(x)+μ T (h(x) (γ1,...,γ 2 l 2 ) T ) λ R m, μ R l, γ R l J ext = J 2014 17 / 119

(2) = (ISS) FB J ext (x, λ, μ, γ) =0 x : J ext x = F x + λt g x λ : g(x) =0 μ : h(x) (γ1,...,γ 2 l 2)T =0 γ : μ i γ i =0 + μt h x =0 n + m +2l n + m +2l KKT : () h(x) 0, μ 0, diag.(μ)h(x) =0 2014 18 / 119

(3) = (ISS) FB Lagrange @J / @x @g / @x g(x) = 0 J x = λ g x J ext x =0, J ext = J + λg J ext / λ =0 J(x) 2014 19 / 119

= (ISS) FB ( T, ): : ẋ = f(x)+g(x)u (Bolza ): J(x(0); u( )) = F (x(t )) + : K(u) 0 R l : x(0) = x 0 x(0) u(t) T 0 L(x, u)dt : 2014 20 / 119

(1) = (ISS) F (x) : F (x + dx) =F (x)+ df dx dx + O(dx2 ) dx dx F (x( )) x + dx x(t)+δx(t) FB ±x(t) x(t) x(t) + ±x(t) t 2014 21 / 119

(2) = (ISS) FB F (x( )) = φ(x(t )) + F (x + δx) F (x) = φ δx(t )+ x(t ) = φ δx(t )+ x(t ) T 0 [ L ẋ δx ( ) T 0 0 L(x(t), ẋ(t), t)dt L L δx + x ẋ δẋdt+ O(δx2 ) ] T T [ L + x d dt 0 ( )] L δx dt + O(δx 2 ) ẋ L φ (x(0), ẋ(0), 0) = 0, ẋ x(t ) + L (x(t ), ẋ(t ),T)=0, ẋ L x d ( ) L =0 (Euler-Lagrange ) dt ẋ 2014 22 / 119

(1) = (ISS) FB ( : Brachistocrone): ( ) A B y(x) (Bernoulli Challenge 1696) y A mg T = a 0 B x 1+y 2 dx min 2gy v = 2gy = 1+y 2 ẋ 1+y dt = 2 2gy dx 2014 23 / 119

(2) = (ISS) Euler-Lagrange : 1+y 2 +2yy =0 (Cycloid) : A FB B 2014 24 / 119

(1) = (ISS) FB p(t) R n, μ(t) R l, γ(t) R l : J ext = F (x(t )) + + + T 0 T 0 T 0 L(x, u)dt p(t) T (f(x)+g(x)u ẋ)dt μ(t) T (K(u) (γ 2 1,...,γ 2 l ) T )dt 2014 25 / 119

(2) = (ISS) FB : ( ) H(x, p, u) =L(x, u)+p T (f(x)+g(x)u) J ext = F (x(t )) + T 0 p T ẋdt + T 0 T 0 H(x, p, u)dt μ(t) T (K(u) diag(γ)γ)dt 2014 26 / 119

(1) = (ISS) : x, p, u, μ, γ x(t) =x (t)+δx(t), μ(t) =μ (t)+δμ(t) p(t) =p (t)+δp(t), γ(t) =γ (t)+δγ(t) u(t) =u (t)+δu(t) Jext FB J ext = F (x (T )+δx(t )) + T 0 T 0 H(x + δx, p + δp, u + δu)dt (p + δp) T (ẋ + δẋ)dt + T 0 ((γ 1 δγ 1 ) 2,...,(γ l δγ l ) 2 ) T }dt (μ + δμ) T {K(u + δu) 2014 27 / 119

(2) = (ISS) FB 1 δx(0) = 0 J ext = Jext + F x (x (T ))δx(t) T ( ) H H H + δx + δp + x p u δu 0 p T (T )δx(t ) + + + 2 T μ T 0 T 0 T 0 K u T 0 u=u ṗ T δxdt δudt {K(u ) diag(γ )γ } T δμdt μ T diag(γ )δγdt + dt x=x,p=p,u=u T 0 ẋ T δpdt 2014 28 / 119

(3) = (ISS) FB [ H(x δp : ẋ,p,u ] T ) = p [ H(x δx : ṗ,p,u ) = x [ F(x δx(t ) : p (T )) (T )= x (T ) : x (0) = x 0 δu : H(x,p,u ) u + μ T K(u ) u =0 δμ : K(u ) diag(γ )γ =0 δγ : μ T diag(γ )=0 ] T ] T 2014 29 / 119

(1) = (ISS) FB x p : : : [ H ẋ = p [ H ṗ = x p(t )= ] T x(0) = x 0 ] T [ F ] T x(t ) 2014 30 / 119

(2) = (ISS) FB u Remark: H K + μt u u =0, K(u) diag(γ)γ =0, μ T diag(γ) =0 x p H(x, p, u) min, subject to K(u) 0 u : u K(u) 0 H Hamiltonian L abnormal multiplier 2014 31 / 119

(1) = (ISS) FB Finite Horizon x(0) = x 0 : 1. u x p u (x, p) 2. : ] T [ H ẋ = p [ H ṗ = x ] T x(0) = x 0, p(t )= [ F ] x(t ) u = u (x, p) 3. u (x, p) 2014 32 / 119

(2) = (ISS) FB 2. 2 u u 1 u 0 ( = ) 2014 33 / 119

(1) = (ISS) FB (u ) Ḣ = H x ẋ + H p ṗ + H u u ( = H x H p T = μ T dk(u) dt + H p H x K i (u) > 0 μ i =0 K i (u) =0dK i /dt =0 t Ḣ =0 T ) μ T K u u () u H u H? 2014 34 / 119

(2) = (ISS) FB x p () u H u H H Infinite Horizon H =0 2014 35 / 119

= (ISS) ( ) H HJ Riccati OFP FB 2014 36 / 119

( ) = (ISS) ( T, ): ( ) H HJ Riccati OFP : ẋ = f(x)+g(x)u (Bolza ): J(x(0); u( )) = F (x(t )) + : K(u) 0 R l : x(0) = x 0 T 0 L(x, u)dt min FB 2014 37 / 119

= (ISS) ( ) H HJ Riccati OFP [0,T] {x ( ),u ( )} [τ,t], x (τ) {ˆx ( ), û ( )} ˆx (t) =x (t), û (t) =u (t), t [τ,t] u*(t) x*( ) b = x*( ) u*(t) b FB 0 T u (t) T t x(0) 2014 38 / 119

= (ISS) ( ) H HJ Riccati OFP FB : u = α(x, t) (T = ) (F (x(t )) = 0) u = α(x) f(0) = 0, L(0, 0) = 0, L(x, u) 0 T = u U(x(0)) = {u( ) : x(0) x(t) 0 } u 2014 39 / 119

= (ISS) ( ) H HJ Riccati OFP FB L(x, u) =L 0 (x)+ 1 2 ut Ru L 0 (x) ( ), R ( ) : 0 H(x, p, u) =p T (f(x)+g(x)u)+l 0 (x)+ 1 2 ut Ru p = V x T, V(0) = 0 H(x, p, u)dt + V (x(0)) = J(x(0); u), u U(x(0)) 2014 40 / 119

H = (ISS) ( ) H HJ Riccati OFP FB u H(x, p, u) =p T f(x)+l 0 (x) 1 2 pt g(x)r 1 g(x) T p + 1 2 (u + R 1 g(x) T p) T R(u + R 1 g(x) T p) ( ) H(x, p, u) u = R 1 g(x) T p H (x, p) =p T f(x)+l 0 (x) 1 2 pt g(x)r 1 g(x) T p H(x, p, u) H (x, p) 2014 41 / 119

Hamilton-Jacobi (1) = (ISS) ( ) H HJ Riccati OFP FB Hamilton-Jacobi (HJ ): ( H x, V x T ) = V x f(x)+l 0(x) 1 2 V x g(x)r 1 g(x) T V x V (x) T =0 p T = V/ x 2014 42 / 119

Hamilton-Jacobi (2) = (ISS) ( ) H HJ Riccati OFP FB V (x(0)) 1 (u + R 1 g(x) T V ) T R(u + R 1 g(x) T V 2 0 x x + V (x(0)) = J(x(0),u), u U(x(0)) J(x(0),u) u = u (x) = R 1 g(x) T V x J V (x(0)) V (x) T T T )dt 2014 43 / 119

(1) = (ISS) ( ) H HJ Riccati OFP V (x) Lyapunov V (x) : u = R 1 g(x) T ( V/ x) T V (x(t)) : V = V x (f(x)+g(x)u )= L(x, u ) = L 0 (x) 1 2 u (x) T Ru (x) 0 FB 2014 44 / 119

(2) = (ISS) ( ) H HJ Riccati OFP FB L(x, u (x)) = 0 u (x) =0and L 0 (x) =0 : L(x, u) L 0 (x) () L 0 (x) V (x) HJ u = u (x) = R 1 g(x) T V/ x T J(x(0); u )=V (x(0)) V (x) 2014 45 / 119

= (ISS) ( ) H HJ Riccati OFP FB 1. u =0 L 0 (x) 2. Hamilton-Jacobi : H (x, V x )=0 V (x) 3. u = u (x) = R 1 g(x) T V x T 2014 46 / 119

(1) = (ISS) ( ) H HJ Riccati OFP : ẋ = Ax + Bu : J = 1 2 : V (x) = 1 2 xt Px, P > 0 Hamilton-Jacobi : 0 x T Qx + u T Ru dt min, 1 2 xt ( PA+ A T P PBR 1 B T P + Q ) x =0 Q > 0,R>0 FB 2014 47 / 119

(2) = (ISS) ( ) H HJ Riccati OFP Riccati : PA+ A T P PBR 1 B T P + Q =0 P (> 0) u = R 1 B T Px (A, B) Riccati FB 2014 48 / 119

(1) = (ISS) ( ) H HJ Riccati OFP FB : J = T 0 L 0 (x)+ 1 2 ut Rudt, R > 0 u: u (x, p) = R 1 g(x) T p ẋ = Ax + Bu + O((x, u) 2 ) J = 1 2 T (A, B), Q>0 0 x T Qx + u T Ru dt + O(x 3 ) : ( ) ( ) [ d x x A BR A dt p H = 1 B T p Q A T ]( ) x p A H 2014 49 / 119

(2) = (ISS) ( ) H HJ Riccati OFP FB : A H A H (λi A H ) A H λ λ ( ) f g =0 λ (f T,g T ) T ( g T,f T )( λi A H )=0 A H λ ( g T,f T ) (H ) n n 2014 50 / 119

(1) = (ISS) ( ) H HJ Riccati OFP n n p(t ) = 0 p x(0) = x 0 x Stable manifold T = p( ) =0 FB Unstable manifold 2014 51 / 119

(2) = (ISS) ( ) H HJ Riccati OFP FB x p T = ω(x) : u = R 1 g(x) T ω(x) : p T = ω(x) ω ω = V x V (x) p =[ V/ x] T 2014 52 / 119

Riccati (1) = (ISS) ( ) H HJ Riccati OFP FB Riccati PA+ A T P PBR 1 B T P + Q =0 ( = ) ( ) : ( ) ( ) [ ]( d x x A BR = A dt p H = 1 B T x p Q A p) T p =[ V/ x] T = Px P 2014 53 / 119

Riccati (2) = (ISS) ( ) H HJ Riccati OFP FB 1. A H : [ ] [ ] S1 S1 A H = Λ S 2 S 2 Λ: n 2. (x T,p T ) T ( ) ( ) [ ] x x S1 = = k p Px x = S 1 k, Px = S 2 k k Px = S 2 S 1 1 x Riccati P = S 2 S 1 1 S 2 2014 54 / 119

OFP (1) = (ISS) ( ) H HJ Riccati OFP FB HJ V = L f V + L g V u L g V u + 1 2 L gv (x)r 1 L g V (x) T û = R 1/2 u, ŷ = R 1/2 L g V (x) T = R 1/2 u (x) 0 V û T ŷ + 1 2ŷT ŷ û ŷ OFP( 1/2) + { u^ u(= u*) x y^ R {1/2 R {1/2 L g V(x) T Plant :, :60, : 2014 55 / 119

OFP (2) = (ISS) ( ) H HJ Riccati OFP FB : 2 (1) 1 ẋ = f(x)+g(x)u u = k(x) L 0 (x) 0 (L 0 (0) = 0) J = (2) 2 0 L 0 (x)+ 1 2 ut Ru dt ẋ = f(x)+ĝ(x)û = f(x)+g(x)r 1/2 û y = R 1/2 k(x) OFP( 1/2) 2014 56 / 119

OFP (3) = (ISS) ( ) H HJ Riccati OFP FB (1) 2 OFP(-1/2) (1) 2 ẋ = f(x) g(x)k(x) y = R 1/2 k(x) ẋ = f(x), y = R 1/2 k(x) (1) (2) (2) (1) (2) Hill=Moylan q(x) S(x) L f S = 1 2 q(x)t q(x)+ 1 2 k(x)t Rk(x) L g S = k(x) T R 1/2 S(x) =V (x), 2L 0 (x) =q(x) T q(x) HJ k(x) =0 L 0 (x) =0 u = k(x) 1 2014 57 / 119

(1) = (ISS) ( ) H HJ Riccati OFP L 0 (x) OFP( 1/2) y u = y L 0 (x) : L 0 (x) FB 2014 58 / 119

(2) = (ISS) ( ) H HJ Riccati OFP FB ( ): clf V (x) u = k 1 (x) = (1/2)R(x) 1 (L g V ) T (R(x) ) V (x) 2 u = k 2 (x) = R(x) 1 (L g V ) T : u = k 1 (x) V = L f V + L g V k 1 = W (x) < 0 (x 0) V = L f V + LĝV û = W (x)+lĝv(û ŷ/2) < ŷ (û T 1 ) (x 0) 2ŷ OFP( 1/2) 2014 59 / 119

(3) = (ISS) ( ) H HJ Riccati OFP FB u = R(x) 1 L G V T (R(x) > 0) Jurdjevic-Quinn Sontag Jurdjevic-Quinn L g V (x) =0 x R(x) 1 R(x) 2014 60 / 119

(4) = (ISS) ( ) H HJ Riccati OFP FB Sontag clf V = 1 { } L f V L f V 2 2 +(L g V L g V T ) 2 < 0 (x 0) L g V (x) =0 Sontag : u = α S(x) =α S (x) c L g V T (c>0) (û ŷ ) u α S (x) c 2014 61 / 119

= (ISS) L 2 = H HJI FB H FB 2014 62 / 119

(1) = (ISS) : L 2 = H HJI FB H : (=H ) H L 2 L 2 FB 2014 63 / 119

(2) = (ISS) L 2 = H HJI FB FB (s) (s) + + + G 0 (s) G 0 (s) { : Δ(s) Δ(jω) <h(ω) h(ω) Δ(s) Δ(s) =0 1+G 0 (jω) >h(ω) 2014 64 / 119

(3) = (ISS) + { (s) L 2 = H HJI FB FB 1 1 + G 0 (s) Δ(jω) 1 1+G 0 (jω) < 1 Δ(jω) < 1+G 0(jω) 1+G 0 (jω) >h(ω) 1 2014 65 / 119

(4) = (ISS) L 2 = H HJI FB FB h(ω) L L 0.040 0.080 0.119-0.593 0.459 0.001-0.001-100 0.0-1.0-0.5 0.0 0.609-0.451 0.5 2-0.032-0.072-0.111 0.163-0.155 0.6 0.21 0.4 0.2-0.2-0.4-0.202-0.6 0.261-0.253 0.317-0.309 0.380-0.372 2014 66 / 119

L 2 = (ISS) L 2 = H HJI FB FB? : ẋ = f(x)+g(x)w z = h(x) x R n :, w R m :, z R p : f, g, h f(0) = 0, h(0) = 0 L 2 : t t 0 z 2 dt γ 2 γ>0 x(t 0 )=0 t t 0 w 2 dt, w L 2 L c, t>t 0 c γ L 2 L 2 γ 2014 67 / 119

(1) = (ISS) L 2 = H HJI FB FB V (x) s(w, z) x(t 0 )=0 t t 0 s(w, z)dt 0, t t 0 γ L 2 V (x) s(w, z) =γ 2 w 2 z 2 2014 68 / 119

(2) = (ISS) L 2 = H HJI FB FB ( ) : ( V s(...)) L f V + L g V w γ 2 w 2 h(x) 2 = γ 2 w T w h(x) T h(x) w L f V +h T h+ 1 4γ 2 L gv L g V T γ 2 (w 1 2γ 2 L gv T ) T (w 1 2γ 2 L gv T ) 0 w =(1/2γ 2 )L g V T () L f V + h T h +(1/4γ 2 )L g V L g V T w L f V + h T h + 1 4γ 2 L gv L g V T 0 = 2014 69 / 119

= = (ISS) L 2 = H HJI FB FB = L f V + h T h + 1 4γ 2 L gv L g V T 0 V (x) γ L 2 1. HJ 2. 2014 70 / 119

(1) = (ISS) L 2 = H HJI FB FB V (x) = V (x) w =0 V = L f V h T h 1 4γ 2 L gv L g V T h(x) T h(x) z = h(x) w =0 w =0 HJ L f V + h T h + 1 4γ 2 L gv L g V T < 0 (x 0) w =0 V < 0 (x 0) 2014 71 / 119

(2) = (ISS) L 2 = H HJI FB FB z = h(x) w =0 HJ (HJ ) V (x) γ L 2 (V (x) ) V (x) Hamiltonian HJ 2014 72 / 119

= (ISS) L 2 = H HJI FB FB ẋ = Ax + Bw, z = Cx γ L 2 : PA+ A T P + CC T + 1 γ 2 PBBT P 0 P γ L 2 : PA+ A T P + CC T + 1 γ 2 PBBT P<0 (x 0) P V (x) =x T Px 2014 73 / 119

H (1) = (ISS) L 2 = H HJI FB FB : ẋ = f(x)+g 1 (x)w + g 2 (x)u z = h(x)+j 1 (x)w + j 2 (x)u x R n :, w R m :, u R l :, z R p : x =0 f(0) = 0, h 1 (0) = 0 : w( ) z( ) L 2 γ (> 0) u 2014 74 / 119

H (2) = (ISS) L 2 = H HJI FB J(x 0,w,u)= 0 z(τ) 2 γ 2 w(τ) 2 dτ, x(0) = x 0 x 0 =0 J w( ) L 2 u = k 2 (x) FB 2014 75 / 119

= (ISS) L 2 = H HJI FB H : u w (= ) w = k 1(x), u = k 2(x) FB J(x 0, w,k 2) J(x 0,k 1,k 2) J(x 0,k 1, u), w; u U(x 0,k 1) k 1, k 2 U(x 0,k 1) w = k 1(x) x 0(t ) u( ) 2014 76 / 119

(1) = (ISS) L 2 = H HJI FB FB : H(x, p, w, u) = h(x)+j 1 (x)w + j 2 (x)u 2 γ 2 w 2 + p(f(x)+g 1 (x)w + g 2 (x)u) (w, u) (w (x, p),u (x, p)) (x, p) H(x, p, w, u (x, p)) H(x, p, w (x, p),u (x, p)) H(x, p, w (x, p),u) : x R 1 (x) =γ 2 I j 1 (x) T j 1 (x) > 0 R 2 (x) =j 2 (x) T j 2 (x) > 0 2014 77 / 119

(2) = (ISS) L 2 = H HJI FB FB w (x, p) = 1 2 Λ 1 {(g T 1 Ξg T 2 )p T +2(j T 1 Ξj T 2 )h} u (x, p) = 1 2 Γ 1 {(Ωg T 1 + g T 2 )p T +2(Ωj T 1 + j T 2 )h} Ξ=j T 1 j 2 R 1 2, Ω=jT 2 j 1 R 1 1, Λ=R 1 +Ξj T 2 j 1 > 0, Γ=R 2 +Ωj T 1 j 2 > 0 H(x, p, w, u) =H (x, p) (w w ) T Λ(w w ) + {u u +Ξ T (w w )} T R 2 {u u +Ξ T (w w )} = H (x, p)+(u u ) T Γ(u u ) {w w +Ω T (u u )} T R 1 {w w +Ω T (u u )} 2014 78 / 119

Hamilton-Jacobi-Isaacs = (ISS) L 2 = H HJI FB FB Hamilton-Jacobi-Isaacs : H H H ( x, V x ( x, V x,w,u ( x, V x,w ) ( x, V x ) ( x, V x =0 )) 0, ),u 0, w u 2014 79 / 119

(1) = (ISS) L 2 = H HJI FB FB : H V (x(t 1 )) V (x 0 )+ V (x(t 1 )) V (x 0 )+ ( x, V ) x,w,u t1 0 t1 0 = dv dt + z 2 γ 2 w 2 z 2 γ 2 w 2 dt 0, z 2 γ 2 w (x, V x ) 2 dt 0, u; w = w (x, V x ) u U(x 0,w (x, V/ x)) V (x) 0 w; u = u (x, V x ) J(x 0,w,u ) V (x 0 ) J(x 0,w,u), w, u U(x 0,w ) 2014 80 / 119

(2) = (ISS) L 2 = H HJI FB u U(x 0,w ) u, w V (x) =J(x, u,w ) k 1(x) =w (x, V x ), J(0,w,u ) 0 k 2(x) =u (x, V x ) FB 2014 81 / 119

(3) = (ISS) L 2 = H HJI FB FB Hamilton-Jacobi-Isaacs V x f (x)+h T (x)q(x)h(x)+ 1 4 V x { (g1 (x) g 2 (x)ξ T (x)) Λ(x) 1 (g T 1 (x) Ξ(x)g T 2 (x)) g 2 (x)r 2 (x) 1 g T 2 (x) } V x T =0 V (0) = 0, V (x) 0 V (x) f = f g 2 R2 1 jt 2 h 1 +(g 1 g 2 Ξ T )Λ 1 (j1 T Ξj2 T )h Q = I j 2 R2 1 jt 2 +(j 1 j 2 Ξ T )Λ 1 (j1 T Ξj2 T ) 0 u = k 2(x) =u (x, V/ x) w z L 2 γ>0 Hamilton-Jacobi- Isaacs V (x) 2014 82 / 119

H (1) = (ISS) L 2 = H HJI FB FB Hamilton-Jacobi-Isaacs Hamilton-Jacobi-Isaacs H (x, V x ) 0, V(0) = 0, V(x) 0 V x (f(x)+g 1(x)w + g 2 (x)u (x, V x )) + z 2 γ 2 w 2 = H (x, V x ) (w w (x, V x ))T R 1 (x)(w w (x, V x )) 0 V (x) γ 2 w 2 z 2 u = k 2 (x) =u (x, V/ x) γ 2 w 2 z 2 V (x) 2014 83 / 119

H (2) = (ISS) L 2 = H HJI FB FB z Hamilton-Jacobi-Isaacs V (x) u = k 2 (x) =u (x, V/ x) γ L 2 w =0 γ L 2 w =0 V (x) z 2 z 0 (t ) z =0 x 0 (t ) V (x) HJI 2014 84 / 119

= (ISS) FB HJ HJ H 2014 85 / 119

(1) = (ISS) FB HJ HJ : L 2? L 2 L 2 Disturbance w Control input u Generalized Plant Evaluated Output z Measurment y State-FB Controller Estimated State x» Observer 2014 86 / 119

(2) = (ISS) FB HJ HJ : ẋ = f(x)+g 1 (x)w + g 2 (x)u z = h 1 (x)+j 11 (x)w + j 12 (x)u y = h 2 (x)+j 21 (x)w + j 22 (x)u x R n :, w R m :, u R l : z R p :, y R q : f(0) = 0, h 1 (0) = 0, h 2 (0) = 0 : : ξ = δ(ξ,y) u = α(ξ) w z L 2 γ 2014 87 / 119

(3) = (ISS) FB HJ HJ : u = k 2 (ξ) ξ = f(ξ)+g 1 (ξ)k 1 (ξ)+g 2 (ξ)k 2 (ξ) + G(ξ)(y h 2 (ξ) j 21 (ξ)k 1 (ξ) j 22 (ξ)k 2 (ξ)) k 1 (ξ) =w (ξ, V (ξ)/ ξ) G( ) j 11 =0, j 22 =0 2014 88 / 119

(1) = (ISS) : FB HJ HJ ξ = f(ξ)+g 2 (ξ)k 2 (ξ)+g(ξ)(y h 2 (ξ)) f(ξ) =f(ξ)+g 1 (ξ)k 1 (ξ) h 2 (ξ) =h 2 (ξ)+j 21 (ξ)k 1 (ξ) 2014 89 / 119

(2) = (ISS) FB HJ HJ (= + + ): ẋ E = f E (x E )+g E (x E )r z = h E1 (x E ) v = h E2 (x E ) x E =(x T,ξ T ) T :, v: u, r = w k 1 (x): w ( ) f(x)+g f E (x E )= 2 (x)k 2 (ξ) f(ξ)+g 2 (ξ)k 2 (ξ)+g(ξ)( h 2 (x) h 2 (ξ)) [ ] g g E (x E )= 1 (x) G(ξ)j 21 (x) h E1 (x E )=h 1 (x)+j 12 (x)k 2 (ξ) h E2 (x E )=k 2 (x) k 2 (ξ) 2014 90 / 119

(1) = (ISS) FB HJ HJ FB Hamilton-Jacobi-Isaacs : V x (f g 2R 1 2 jt 12h 1 )+h T 1 (I j 12 R 1 2 jt 12)h 1 + 1 4 V x { 1 γ 2 g 1g T 1 g 2 R 1 2 gt 2 } V x T 0 V (x) u = k 1 (ξ) V x (f(x)+g 1(x)w + g 2 (x)u)+ z 2 γ 2 w 2 = H(x, V x,w,u)=h (x, V x )+ j 12(x)v 2 γ 2 r 2 j 12 (x)v 2 γ 2 r 2 ( ) 2014 91 / 119

(2) = (ISS) FB HJ HJ r j 12 (x)v L 2 γ : W x E (f E (x E )+g E (x E )r) γ 2 r 2 j 12 (x)v 2 2n Hamilton-Jacobi : W f E (x E )+h T x E2(x E )R 2 (x)h E2 (x E ) E + 1 W 4γ 2 g E (x E )g x E(x T E ) W T E x E W (x E ) 0, W(0) = 0 0 G( ) 2014 92 / 119

(1) = (ISS) : FB HJ HJ : U(x E )=W (x E )+V (x) U x E (f E (x E )+g E (x E )r) γ 2 w 2 + z 2 U(x E ) γ 2 w 2 z 2 = γ L 2 2014 93 / 119

(2) = (ISS) FB HJ HJ : 1. FB, Hamilton-Jacobi-Isaacs V (x) 2. W (x E ), G(ξ) 2n Hamilton-Jacobi 3. ξ = f(ξ) G(ξ) h 2 (ξ) ξ =0 γ L 2 w =0 x E =(0, 0) T V (x) 2014 94 / 119

(3) = (ISS) FB HJ HJ : γ L 2 w =0 w =0 du(x E (t)) dt z 2 = h 1 (x)+j 12 (x)k 2 (ξ) 2 U 0 Ω Ω z 0 Ω x 0 (t ) j 12 (x) Ω h 1 (x)+ j 12 (x)k 2 (ξ) =0(x 0) k 2 (ξ(t)) 0 (t ) Ω ξ = f(ξ) G(ξ) h 2 (ξ) 3 Ω ξ 0 (t ) 2014 95 / 119

Hamilton-Jacobi (1) = (ISS) Hamilton-Jacobi 2n Hamilton-Jacobi G( ) FB HJ HJ W (x E )=Q(x ξ) Q(0) = 0, Q(e) 0 e = x ξ Q e ( f(x) f(ξ)+(g 2 (x) g 2 (ξ))k 2 (ξ) G(ξ)( h 2 (x) h 2 (ξ))) +(k 2 (x) k 2 (ξ)) T R 2 (x)(k 2 (x) k 2 (ξ)) + 1 Q 4γ 2 e (g 1(x) j21(x)g T T (ξ)) (g 1 (x) j21(x)g T T (ξ)) T Q T 0 e 2014 96 / 119

Hamilton-Jacobi (2) = (ISS) FB HJ HJ x = ξ + e K(e, ξ) Q/ e e=0 =0 K K(0,ξ)=0, e =0 e=0 K(e, ξ) (0, 0) Hessian ( 2 Q f ) e 2 e=0 x (0) G(0) h 2 x (0) + k T 2 (0)R 2 (0) k 2 x x (0) + 1 2 Q 4γ 2 e 2 (g 1 (0) j21(0)g T T (0)) e=0 (g 1 (0) j21(0)g T T (0)) T 2 T Q e 2 e=0 2014 97 / 119

Hamilton-Jacobi (3) = (ISS) FB HJ HJ 1. FB Hamilton-Jacobi-Isaacs C 3 V (x) 2. ɛ>0 S(e) = Q e ( f(e) G(e) h 2 (e)) + k2 T (e)r 2 (e)k 2 (e) + 1 Q 4γ 2 e (g 1(e) G(e)j 21 (e)) (g 1 (e) G(e)j 21 (e)) T Q e ɛe T e<0, (e 0) T C 3 Q(e) ( 0) W (x E )=Q(x ξ) 2n 2, 3 2014 98 / 119

Hamilton-Jacobi (4) = (ISS) FB HJ HJ 2n Hamilton-Jacobi K(0,ξ)=0, ( K/ e) e=0 =0 R(e, ξ) K(e, ξ) =e T R(e, ξ)e R(e, ξ) < 0 2n Hamilton-Jacobi R(0, 0) (0, 0) K(e, ξ) e Hessian n Hamilton-Jacobi e =0 Hessian 2 S/ e 2 (0, 0) K(e, ξ) e Hessian R(e, ξ) K(e, ξ) e =0 2n Hamilton-Jacobi Hamilton-Jacobi Q e ( f(ξ) G(ξ) h 2 (ξ)) < 0, (ξ 0) e=ξ Q Ω Lyapunov 2014 99 / 119

(1) = (ISS) FB HJ HJ HJ S G( ) : R 3 (x) =j 21 (x)j T 21(x) ( ) G(x) : Q(x) x f(x)+k T 2 (x)r 2 (x)k 2 (x)+ 1 Q(x) 4γ 2 x g 1(x)g1 T (x) Q(x) T x + 1 4γ 2 (λ(x) BT (x)r3 1 (x))r 3(x)(λ(x) B T (x)r3 1 (x))t 1 (x)b(x) < 0, (x 0) 4γ 2 BT (x)r3 1 e x λ(x) = Q(x) x G(x), B(x) =2γ2 h2 (x)+j 21 (x)g1 T (x) Q(x) T x 2014 100 / 119

(2) = (ISS) FB HJ HJ G(x) HJ S(x) λ(x) B T (x)r 1 3 (x) = Q(x) x G(x) (2γ 2 ht 2 (x)+ Q(x) x G(x) ( ) G(x) Q x L(x) = h T 2 (x) g 1(x)j21(x))R T 3 1 (x) =0 n q L(x) 2014 101 / 119

(3) = (ISS) FB HJ HJ Q(x)/ x Q(x) x = x T M(x) h T 2 (x) =x T N(x) G(x): M(x) x =0 L(x) =M 1 (x)n(x) G(x) =(2γ 2 L(x)+g 1 (x)j T 21(x))R 1 3 (x) G(x) 2014 102 / 119

Hamilton-Jacobi (1) = (ISS) FB HJ HJ G(x) Hamilton-Jacobi : Q(x) x f(x)+k2 T (x)r 2 (x)k 2 (x)+ 1 Q(x) 4γ 2 x 1 4γ 2 BT (x)r3 1 (x)b(x) < 0, (x 0) Q(x) x ( f(x) g 1 (x)j T 21(x)R 1 3 (x) h 2 (x)) + k T 2 (x)r 2 (x)k 2 (x) γ 2 ht 2 (x)r 1 3 (x) h 2 (x) + 1 Q(x) 4γ 2 x < 0 (x 0) g 1(x)g1 T (x) Q(x) T x g 1(x)(I m j T 21(x)R 1 3 (x)j 21(x))g T 1 (x) Q(x) x T 2014 103 / 119

Hamilton-Jacobi (2) = (ISS) FB HJ HJ Hamilton-Jacobi Q( ) M(x) G(x) (Q( ),G(x)) n Hamilton- Jacobi 2014 104 / 119

= (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 2014 105 / 119

() = (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 Strict feedback system Hessian 2014 106 / 119

Strict feedback system = (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 Strict feedback system: ẋ 1 = f 1 (x 1 )+g 1 (x 1 )x 2 ẋ 2 = f 2 (x 1,x 2 )+g 2 (x 1,x 2 )x 3 ẋ 3 = f 3 (x 1,x 2,x 3 )+g 3 (x 1,x 2,x 3 )x 4. ẋ s 1 = f s 1 (x 1,...,x s 1 )+g s 1 (x 1,...,x s 1 )x n ẋ s = f s (x)+g s (x)u Σ k x k+1 Σ n (=) u dim x 1 dim x 2 =dimx 3 = =dimx s. g 2 (x 1,x 2 ),...,g s (x) 2014 107 / 119

Σ 1 = (ISS) Σ 1 FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 ẋ 1 = f 1 (x 1 )+g 1 (x 1 )v 1 (x 2 v 1 ) : Σ 1 v 1 = α 1 (x 1 ) V 1 (x 1 ) V 1 x 1 (f 1 + g 2 α 1 ) < 0 (x 1 0) 2014 108 / 119

Σ 2 (1) = (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 Σ 2 : ẋ 1 = f 1 (x 1 )+g 1 (x 1 )x 2 ẋ 2 = f 2 (x 1,x 2 )+g 2 (x 1,x 2 )v 2 Σ 2 Σ 1 v 1 x 2 z 2 = x 2 α 1 (x 1 ) V 2 (x 1,z 2 )=V 1 (x 1 )+ γ 2 2 zt 2 z 2 2014 109 / 119

Σ 2 (2) : V 2 = V 1 {f 1 + g 1 α 1 } + V 1 g 1 z 2 + γ 2 z2 T x 1 x 1 { = V 1 x 1 {f 1 + g 1 α 1 } + z T 2 g T 1 [ V1 x 1 { (f 2 + g 2 v 2 ) α 1 } (f 1 + g 1 x 2 ) x 1 ] T + γ 2 (f 2 + g 2 v 2 ) γ 2 α 1 x 1 (f 1 + g 1 x 2 ) < 0 (x 1 0) k 2 z 2 } g T 1 [ V1 x 1 ] T + γ 2 (f 2 + g 2 v 2 ) γ 2 α 1 x 1 (f 1 + g 1 x 2 )= k 2 z 2 = k 2 (x 2 + α(x 1 )) v 2 Σ 2 v 2 = α 2 (x 1,x 2 ) 2014 110 / 119

Σ 3 (1) = (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 Σ 3 : : : ẋ 1 = f 1 (x 1 )+g 1 (x 1 )x 2 ẋ 2 = f 2 (x 1,x 2 )+g 2 (x 1,x 2 )x 3 ẋ 3 = f 2 (x 1,x 2,x 3 )+g 2 (x 1,x 2,x 3 )v 3 z 3 = x 3 α 2 (x 1,x 2 ) V 3 (x 1,z 2,z 3 )=V 2 (x 1,z 2 )+ γ 3 2 zt 3 z 3 ( ) ( ) ( x1 f1 + g x 2 =, f2 (x 1,x 2 )= 1 x 1 0, ḡ 2 (x 1,x2) = g2) x 2 f 2 2014 111 / 119

Σ 3 (2) : V 2 = V 2 { x f 2 +ḡ 2 α 2 } + V 2 ḡ 2 z 3 + γ 3 z3 T 2 x 2 { = V 2 x 2 { f 2 +ḡ 2 α 2 } + z T 3 ḡ T 2 [ V2 x 2 { (f 3 + g 3 v 3 ) α 2 } ( x f 2 +ḡ 2 x 3 ) 2 ] T + γ 3 (f 3 + g 3 v 3 ) γ 3 α 2 x 2 ( f 2 +ḡ 2 x 3 ) < 0 ( x 2 0) k 3 z 3 } ḡ T 2 [ V2 x 2 ] T + γ 3 (f 3 + g 3 v 3 ) γ 3 α 2 x 2 ( f 2 +ḡ 2 x 3 )= k 3 z 3 = k 3 (x 3 + α 2 ) v 3 Σ 3 v 3 = α 3 (x 1,x 2,x 3 ) 2014 112 / 119

= (ISS) FB St.fdbk.sys. Σ 1 Σ 2 Σ 3 V 1 Hessian V s Hessian Σ 1 α 1 (x 1 ) Σ 1 u = α s (x) 2014 113 / 119

= (ISS) FB ( ) 1 ISS 2014 114 / 119

( ) = (ISS) Vidyasagar : FB ( ) 1 ISS ẋ = f(x) ż = g(z)+γ(x, z)x ẋ = f(x) ż = g(z) γ(x, z) ẋ = f(x), ż = g(z) 2014 115 / 119

= (ISS) System 1 System 2 FB ( ) 1 ISS ẋ = f(x) ż = g(z)+γ(x, z)x 1 2 ISS 2014 116 / 119

( ) = (ISS) FB ( ) 1 ISS : Lyapunov V 1 (x), V 2 (z) V 1 = L f V 1 c( x ) V 2 = L g V 2 + L γ V 2 x a( z )+b( x ) a, b, c K K k 1, k 2, k 3, k 4 k 1 ( x ) V 1 (x) k 2 ( x ), k 3 ( z ) V 2 (z) k 4 ( z ) K χ a 1 b( x ) <χ( x ), z 0 Lyapunov W = F (V 1 )+G(V 2 )+V 1 F (v) = v 0 b k 1 2 (s)ds, G(v) = v 0 c χ 1 k 1 4 (s)ds 2014 117 / 119

1 = (ISS) FB ( ) 1 ISS : ẋ = f(x) ż = g(z)+γ(x, z)x 1. ẋ = f(x) α( ), M 0 ( ) K, k 0 > 0 x(t) α(m 0 (x(0)) exp( k 0 t)) 2. 2 iiss iiss V 2 (z) V 2 (x, z) a( z )+b( x ), a( ) :, b( ) class-k 3. b(s) : 1 0 b(α(s)) ds < s 1 (α =Id) 2 x 2014 118 / 119

ISS = (ISS) w 1 System1 x 1 = f 1 (x 1, x 2, w 1 ) FB ( ) 1 ISS 1,2 ISS K χ 1, χ 1w, χ 2, χ 2w KL β 1, β 2 x 2 x 1 System2 x 2 = f 2 (x 2, x 1, w 2 ) x 1 (t) β 1 (x 1 (0),t)+χ 1 (sup x 2 )+χ 1w (sup w 1 ) x 2 (t) β 2 (x 2 (0),t)+χ 2 (sup x 1 )+χ 2w (sup w 2 ) K η 1 ( ), η 2 ( ) (Id + η 1 ) χ 1 (Id + η 2 ) χ 2 (s) s (Id + η 2 ) χ 2 (Id + η 1 ) χ 1 (s) s s>0 w 2 w 1, w 2 ISS ISS 1 ISS [1] Z.P. Jiang, A.R. Teel, and L. Praly: Small-gain theorem for ISS systems and applications, Mathe. Contr. Signals and Syst., 7, 95/120 (1994) 2014 119 / 119